This section is devoted to prove Theorem REF . Recall Theorem REF  concludes that the limit of \(max_{\mathcal {C}_{t}}|\nabla ^{m}A|\)  is 0 for all \(m=0,1,\cdots \)  if the curve shortening flow \(\mathcal {C}_{t}\)  in the closed manifold \(\tilde{M}\)  exists for all \(t\in [0, \infty )\)  and the length of \(\mathcal {C}_{t}\)  converges to a positive number. The idea of its proof originated from Section 7 in ([1]}). We suppose the length of \(\mathcal {C}_{t}\)  satisfies \(l_{\mathcal {C}_{t}}\ge l_{\infty }>0\) . First, let's prove a technique lemma.
